Abstract : The Conditional Tail Expectation is an indicator of tail behaviour that has recently gained traction in actuarial and financial applications. Contrary to the quantile or Value-at-Risk, it takes into account the frequency of a tail event together with the probabilistic behaviour of the variable of interest on this event. However, the asymptotic normality of the empirical Conditional Tail Expectation estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in heavy-tailed models which constitute the favoured class of models in actuarial and financial applications. One possible solution in very heavy-tailed models where this assumption fails could be to use the more robust Median Shortfall, but this quantity is actually just a quantile, which therefore only gives information about the frequency of a tail event and not about its typical magnitude. We construct a synthetic class of tail L p −medians, which encompasses the Median Shortfall (for p = 1) and Conditional Tail Expectation (for p = 2). We show that, for 1 < p < 2, a tail L p −median always takes into account both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator, along with another technique, to proper extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on a set of real fire insurance data showing evidence of a very heavy right tail.